Question

In the adjoining figure, TP and TQ are tangents drawn to a circle with centre O. If ∠OPQ = 15° and ∠PTQ = θ, then find the value of sin 2θ.

Answer 1

Given, ∠OPQ = 15°, ∠PTQ = θ

The angle between the tangent and the radius at the point of contact is 90°.

∠1 + 15° = 90°

∠1 = 90° − 15

∠1 = 75°

∠2 = 75°

∠1 = ∠2

∵ TP = PQ   ....(Angles opposite to equal sides are equal.)

In ΔPTQ

∠1 + ∠2 + θ = 180°

75° + 75° + θ = 180°

θ = 180° – 150°

θ = 30°

∴ Sin 2θ

= sin 2 × 30°

= sin 60°

= `sqrt3/2`